To determine which additional operation should be included in the expression \(54x + 6 + ________ \) to make it equivalent to \(12x + 6\), we want to set up the following equation:
\[ 54x + 6 + \text{(additional term)} = 12x + 6 \]
First, we can ignore the \(+6\) part of both sides, since they cancel out:
\[ 54x + \text{(additional term)} = 12x \]
Now, to isolate the additional term, we rearrange the equation:
\[ \text{(additional term)} = 12x - 54x \] \[ \text{(additional term)} = -42x \]
Now, let's evaluate the provided options:
- \( -2x \)
- \( -x \)
- \( \frac{3}{4}x \)
- \( -\frac{34}{x} \)
None of the options directly provide a term of \(-42x\). Let's analyze further:
It appears there might have been a misunderstanding with the question. It seems more plausible that the intention was to simplify the expression using one of the provided options.
Thus, if we specifically select for any of those modifications.
If we want to make \(54x + 6\) decrement towards \(12x + 6\) we can reaffirm to consider the closest operation accounting \(54x + 6\).
A logical deduction suggests calculating \(−42x\) by applying:
- A viably reasonable approximation would yield a negative factor near to \(-34x\) but falls short. Thus evaluating further we spot for \(−34x\) it bears examining that:
\[ 54x - 34x + 6 = 20x + 6 \]
Thus it remains unnecessary while fitting: Option (4) would bring closer to \(12x+ 6\) but always fails to precisely closing.
In sum, realizing none of the provided options directly yield the necessary output, you might want to double check parameters or ensure all potential solutions are evaluated!
Ultimately identifying any selectable outcome must consider impact closely to \(12x\).
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